AMIYATOSH PURNANANDAM ET AL. 292.2.2 Economic motivation
An unacceptable portfolio may initially be chosen by a firm which believes it
has superior information or investment skill. Moreover, additional riskfree
capital does not permit a firm to exhibit investment ability or skill. Provided
firms pursue excess economic rents and fail to maintain well diversified
portfolios, a coherent risk measure may overestimate portfolio risk. Funda-
mentally, a tradeoff exists between preventing insolvency and maximizing
the expected value of the portfolio. This tension motivates our risk manage-
ment framework since the firm is able to maintain an acceptable portfolio
as close as possible to their original positions while complying with the
external regulator.
To enhance the motivation behind our risk measure, we introduce a
non-negative functionR(η)≥0 to determine the aggregate desirability of
a portfolio. This function is only intended as an example to illustrate the
desirability of rebalancing a portfolio in comparison to the addition of risk-
free capital. Therefore, neither the functional form nor the exact specification
ofR(η) are necessary for our analysis. Since the selection criteria and per-
ceived desirability of individual assets are highly variable across firms, very
little structure is imposed onR(η). For illustration, we assume:
R(η)=∑N
i= 0 ηi·ci
∑N
i= 0 ηi(2.2)whereciimplicitly denotes a ranking of the assets. For example,cimay
represent numerical weightings associated withstrong outperform,weak out-
perform,orholdamong other possibilities. Equation (2.2) allows several
variables, including expected returns and variances, to influence a portfo-
lio’s desirability. However, covariances are not considered in equation (2.2)
as diversification is reserved for our subsequent discussion of the proposed
risk measure.
Regardless of the exact functional form forR(η), thecielements may be
derived from an infinite number of scenarios, not only those evaluated by the
regulator. Indeed, the regulator is primarily concerned with a small subset
ofextremescenarios. In contrast, the firm’s investment criteria is comprised
of more frequently occurring scenarios. This disparity reflects the diverging
interestsoftheregulatorandfirmwhichourproposedriskmeasureattempts
to bridge.
In the absence of portfolio rebalancing, define the amount of additional
riskfree asset required to ensure the portfolioηbecomes acceptable asα≥0.
This quantity equals
α=inf{γ:η+γηc∈Aη}
(2.3)
=−min{0,Pη}and depends onηbut is written asαrather thanα(η) for notational simplicity.